Σ2 Induction and Infinite Injury Priority Arguments, Part Iii: Prompt Sets, Minimal Pairs and Shoenfield’s Conjecture

Priority constructions are the trademark of theorems on the recursively enumerable Turing degrees. By their combinatorial patterns, they are naturally identified as finite injury, infinite injury, and so forth. Following Chong and Yang [4], [3], we analyze the complexity of infinite injury arguments and pinpoint exactly the position of their degree-theoretic applications within the hierarchy of fragments of Peano arithmetic (cf. Chong and Yang [5] for a discussion of the issues and motivation behind such studies). Finite injury priority constructions fall essentially into two types: the Friedberg-Mučnik type and the Sacks splitting type. For the former, Chong and Mourad [2] show that even though such constructions cannot be carried out without Σ1 induction, Σ1 bounding is a sufficiently strong theory to establish the existence of a pair of incomparable recursively enumerable degrees. For the latter, the results of Mytilinaios [12] and Mourad [11] together imply that the Sacks splitting theorem is equivalent to Σ1 induction over the base theory of Σ1 bounding. Infinite injury constructions, by contrast, are more varied and harder to categorize. Results to-date show that the existence of a high recursively enumerable degree is equivalent to Σ2 induction over the base theory of Σ2 bounding [4], and that the Density Theorem is provable under Σ2 bounding [7] (note that the density theorem fails in all models of Σ1 bounding in which Σ1 induction fails, by a result of Mourad [11]). Our intuition suggests that certain Σ2 properties are necessary for infinite injury arguments to carry through (although, again, there are special models satisfying Σ1